reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve GR,R for RestFunc of REAL-NS n;
reserve DFG,L for LinearFunc of REAL-NS n;

theorem
for f be PartFunc of REAL,REAL n holds
f is_differentiable_on X iff
(for i be Element of NAT st 1<=i & i <=n holds
      Proj(i,n)*f is_differentiable_on X)
proof
  let f be PartFunc of REAL,REAL n;
  thus f is_differentiable_on X implies
  (for i be Element of NAT st 1<=i & i <=n holds
  Proj(i,n)*f is_differentiable_on X) by Th29;
  assume A1: for i be Element of NAT st 1<=i & i <=n holds
  Proj(i,n)*f is_differentiable_on X;
  reconsider g=f as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
  for i be Element of NAT st 1<=i & i <=n holds
  Proj(i,n)*g is_differentiable_on X by A1;
  then
A2: g is_differentiable_on X by Th30;
    then
A3: X is open Subset of REAL by NDIFF_3:9,11;
    then
A4: X c= dom f by A2,NDIFF_3:10;
  for x st x in X holds f is_differentiable_in x by A3,A2,NDIFF_3:10;
 hence thesis by A3,A4,Th5;
end;
